Multiplying Monomials
Learning Outcomes
- Use the power and product properties of exponents to multiply monomials
- Use the power and product properties of exponents to simplify monomials
We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
Properties of Exponents
If [latex]a,b[/latex] are real numbers and [latex]m,n[/latex] are whole numbers, then
[latex-display]\begin{array}{cccc}\text{Product Property}\hfill & & & \hfill {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \text{Power Property}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \text{Product to a Power Property}\hfill & & & \hfill {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \end{array}[/latex-display]
example
Simplify: [latex]{\left({x}^{2}\right)}^{6}{\left({x}^{5}\right)}^{4}[/latex].
Solution
|
[latex]{\left({x}^{2}\right)}^{6}{\left({x}^{5}\right)}^{4}[/latex] |
| Use the Power Property. |
[latex]{x}^{12}\cdot {x}^{20}[/latex] |
| Add the exponents. |
[latex]{x}^{32}[/latex] |
try it
[ohm_question]146171[/ohm_question]
example
Simplify: [latex]{\left(-7{x}^{3}{y}^{4}\right)}^{2}[/latex].
Answer:
Solution
|
[latex]{\left(-7{x}^{3}{y}^{4}\right)}^{2}[/latex] |
| Take each factor to the second power. |
[latex]{\left(-7\right)}^{2}{\left({x}^{3}\right)}^{2}{\left({y}^{4}\right)}^{2}[/latex] |
| Use the Power Property. |
[latex]49{x}^{6}{y}^{8}[/latex] |
try it
[ohm_question]146174[/ohm_question]
example
Simplify: [latex]{\left(6n\right)}^{2}\left(4{n}^{3}\right)[/latex].
Answer:
Solution
|
[latex]{\left(6n\right)}^{2}\left(4{n}^{3}\right)[/latex] |
| Raise [latex]6n[/latex] to the second power. |
[latex]{6}^{2}{n}^{2}\cdot 4{n}^{3}[/latex] |
| Simplify. |
[latex]36{n}^{2}\cdot 4{n}^{3}[/latex] |
| Use the Commutative Property. |
[latex]36\cdot 4\cdot {n}^{2}\cdot {n}^{3}[/latex] |
| Multiply the constants and add the exponents. |
[latex]144{n}^{5}[/latex] |
Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.
try it
[ohm_question]146177[/ohm_question]
example
Simplify: [latex]{\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}[/latex].
Answer:
Solution
|
[latex]{\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}[/latex] |
| Use the Power of a Product Property. |
[latex]{3}^{4}{\left({p}^{2}\right)}^{4}{q}^{4}\cdot {2}^{3}{p}^{3}{\left({q}^{2}\right)}^{3}[/latex] |
| Use the Power Property. |
[latex]81{p}^{8}{q}^{4}\cdot 8{p}^{3}{q}^{6}[/latex] |
| Use the Commutative Property. |
[latex]81\cdot 8\cdot {p}^{8}\cdot {p}^{3}\cdot {q}^{4}\cdot {q}^{6}[/latex] |
| Multiply the constants and add the exponents for
each variable. |
[latex]648{p}^{11}{q}^{10}[/latex] |
try it
[ohm_question]146179[/ohm_question]
Multiply Monomials
Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.
example
Multiply: [latex]\left(4{x}^{2}\right)\left(-5{x}^{3}\right)[/latex].
Answer:
Solution
|
[latex]\left(4{x}^{2}\right)\left(-5{x}^{3}\right)[/latex] |
| Use the Commutative Property to rearrange the factors. |
[latex]4\cdot \left(-5\right)\cdot {x}^{2}\cdot {x}^{3}[/latex] |
| Multiply. |
[latex]-20{x}^{5}[/latex] |
try it
[ohm_question]146195[/ohm_question]
example
Multiply: [latex]\left(\frac{3}{4}{c}^{3}d\right)\left(12c{d}^{2}\right)[/latex].
Answer:
Solution
|
[latex]\left(\frac{3}{4}{c}^{3}d\right)\left(12c{d}^{2}\right)[/latex] |
| Use the Commutative Property to rearrange
the factors. |
[latex]\frac{3}{4}\cdot 12\cdot {c}^{3}\cdot c\cdot d\cdot {d}^{2}[/latex] |
| Multiply. |
[latex]9{c}^{4}{d}^{3}[/latex] |
try it
[ohm_question]146196[/ohm_question]
For more examples of how to use the power and product rules of exponents to simplify and multiply monomials, watch the following video.
https://youtu.be/E_D8PO1G7gULicenses & Attributions
CC licensed content, Original
- Question ID 146196, 146148, 146197. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 2: Exponent Properties (Product, Power Properties). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
